Three-dimensional Incompressible Boundary Layer Research Articles

Analysis and Computation of Extremum Points with Application to Boundary-Layer Stability

A method for analysis and computation of derivatives and extremum points of variable-coefficients differential eigenvalue problems is presented. The method utilizes the orthogonality of the adjoint eigenfunctions to the homogenous part of the once or more differentiated problem to derive an analytical expression for the rate of change of eigenvalue with respect to a free parameter. The extremum point can be analyzed and computed by setting and driving, respectively, the first rate of change of the eigenvalue with respect to the free parameters to zero. Higher order derivatives can be computed by solving, sequentially, sets of inhomogeneous two-point boundary value problems. The method is applied to analyze and compute the most amplified inviscid instability wave in two-dimensional compressible boundary layers and the most amplified viscous instability wave in three-dimensional incompressible boundary layers. It is shown analytically that while the most-amplified spatial instability wave in two-dimensional incompressible boundary layer is two dimensional, the corresponding most amplified wave in three-dimensional boundary layer is generally oblique. It is also shown analytically that the most-amplified disturbance in three-dimensional boundary layer is generally a traveling disturbance. Furthermore, it is shown analytically that the inviscid growth rate is an extremum point.

Investigating flow topologies in boundary layer transition using direct numerical simulation

A numerical method developed for simulating three-dimensional incompressible boundary layer flow is presented. K-type transition up to the two-spike stage is simulated, and flow topologies at various stages of transition are determined. Comparison with flow topologies from other simulations of turbulent and transitioning flows is made.

The effects of suction on the nonlinear stability of a three-dimensional compressible boundary layer

This study follows on from work by Bassom & Seddougui (1992, Proc. R. Soc. Lond . A 436 , 405–15) on the effects of suction on the nonlinear stability of the three-dimensional incompressible boundary layer induced by a rotating disc. This flow has two types of stationary instability, one corresponds to the upper-branch inviscid mode and the other to the lower-branch viscous mode. This latter instability is characterized by an effective velocity profile which has a zero shear stress at the wall and, as in Bassom & Seddougui (1992), it is on this mode that interest is focused here. The effect of suction on the compressible flow, and its subsequent instability, is shown to be significant.

Secondary instability in rotating-disk flow

Primary instability of the three-dimensional boundary layer on a rotating disk introduces periodic modulation of the mean flow in the form of stationary crossflow vortices. Here we study the stability of this modulated mean flow with respect to secondary disturbances. These secondary disturbances are found to have quite large growth rates compared to primary disturbances. Both fundamental and subharmonic resonance cases are considered and their corresponding results indicate that the growth rate and the frequency of the secondary instability are insensitive to the exact nature of the resonance condition. The threshold primary stationary crossflow vortex amplitude for secondary instability found in this three-dimensional incompressible boundary layer is significantly larger than that for a two-dimensional boundary layer which is subjected to Tollmien–Schlichting instability. The secondary instability results in a pair of travelling counter-rotating vortices, tilted up and oriented at an angle to the primary stationary crossflow vortices. The computed velocity signals and flow visualization, evaluated based on this secondary disturbance structure, are compared with experimental results.

Solution of three-dimensional boundary layer equations by a discontinuous finite element method, Part II: Implementation and numerical results

We present, in this second paper, a generalization of the numerical method developed in Part I to the nonlinear system of three-dimensional incompressible boundary layer equations. A nonlinear block Gauss-Seidel iterative solver, allowing a solution strategy by progressing on the surface on which lies the boundary layer, is described. Numerical tests are given showing the efficiency of the method.

Boundary-layer transition on a rotating cone in still fluid

A linear stability analysis and experiments were carried out for the laminar–turbulent transition of three-dimensional incompressible boundary layers induced on the surface of a cone rotating around the axis of symmetry with constant angular speed in still fluid. Five cones having total angle of 30°–150° were tested. The results show that the critical and transition Reynolds numbers, the direction of spiral vortices appearing in the transition region and their number on a cone increase as the cone angle is increased, and they tend to the values for the case of a rotating disk. Flow visualizations were made for the transitional process and also for cross-sectional flows of spiral vortices.

Optimal coordinates for three-dimensional incompressible laminar boundary layers

This paper summarizes the basic theory of optimal coordinates for the three-dimensional incompressible boundary layers. Though in essence the results obtained in this paper form an extension of Kaplun's work in two dimensions, the additional considerations due to three-dimensionality have extended the scope and applicability of the theory of optimal coordinates to more complicated situations.

Approximate solution of the problem of the three-dimensional boundary layer in an incompressible fluid

A method of successive approximations is proposed for the solution of the equations of the three-dimensional incompressible boundary layer on bodies of arbitrary shape. A coordinate system connected with the streamlines of the external nonviscous flow is used. It is assumed that the velocity across the external streamlines is small. When the intensity of secondary flow is low the equations describing the boundary layer in an incompressible fluid are reduced to a form analogous to the equations for the boundary layer on axially symmetrical bodies. An approximate analytical solution is obtained for the velocity and for the friction in the form of equations which can be used for any problems of a three-dimensional incompressible boundary layer. The method developed was applied to the problem of the three-dimensional boundary layer at a plate with a cylindrical obstacle in the presence of a slip angle.

A new systematic formalism for similarity analysis

A new systematic formalism is presented for similarity analysis—i.e., for reducing the number of independent variables in systems consisting of partial differential equations and a set of auxiliary conditions. The formalism is a specialization of previous group theory techniques developed by the authors. Concurrent with its presentation, and to illustrate that it is particularly well suited for practical similarity analyses, the formalism is applied to certain three-dimensional incompressible boundary layer flows; and a variety of results are elicited which exhibit somewhat greater generality than any previously reported for the class of flows.